Derivative self-interactions for a massive vector field

Abstract: In this work we revisit the construction of theories for a massive vector field with derivative self-interactions such that only the 3 desired polarizations corresponding to a Proca field propagate. We start from the decoupling limit by constructing healthy interactions containing second derivatives of the Stueckelberg field with itself and also with the transverse modes. The resulting interactions can then be straightforwardly generalized beyond the decoupling limit. We then proceed to a systematic construction of the interactions by using the Levi-Civita tensors. Both approaches lead to a finite family of allowed derivative self-interactions for the Proca field. This construction allows us to show that some higher order terms recently introduced as new interactions trivialize in 4 dimensions by virtue of the Cayley-Hamilton theorem. Moreover, we discuss how the resulting derivative interactions can be written in a compact determinantal form, which can also be regarded as a generalization of the Born-Infeld lagrangian for electromagnetism. Finally, we generalize our results for a curved background and give the necessary non-minimal couplings guaranteeing that no additional polarizations propagate even in the presence of gravity.

• The paper introduces a systematic formulation for derivative self-interactions of a Proca field, ensuring only the three desired polarizations propagate.
• It employs the Levi-Civita tensor and decoupling limit to derive a finite series of interactions that avoid Ostrogradski instabilities.
• The work extends these constructions to curved spacetime by identifying necessary counter-terms that maintain theoretical consistency.

Overview of "Derivative Self-interactions for a Massive Vector Field"

The paper authored by José Beltrán Jiménez and Lavinia Heisenberg provides a comprehensive study on constructing theoretical frameworks for a massive vector field that inherently includes derivative self-interactions while maintaining the propagation of only the three desired polarizations characteristic of a Proca field. This investigation elaborates on methodologies to ensure healthy interactions involving second derivatives of the Stueckelberg and transverse modes. A methodical approach is employed through the use of Levi-Civita tensors to derive a finite series of permissible derivative self-interactions, contributing to the broader understanding and application of Proca fields.

Decoupling Limit and Theoretical Construction

Starting from the decoupling limit, the paper details how certain interactions can be constructed without introducing undesired degrees of freedom. By examining higher derivatives of the Stueckelberg field, the authors emphasize the necessity of maintaining second-order field equations to avoid Ostrogradski instabilities, thus ensuring the theoretical consistency and stability of such fields. The investigation demonstrates that interactions can be constructed as a generalization of Galileon-like theories for vectors using the Levi-Civita tensor, ultimately leading to a compact determinantal representation.

Systematic Construction via Levi-Civita Tensors

By employing Levi-Civita tensors, the authors methodically derive vector field interactions that extend the conceptual framework of Proca fields. The interactions for the massive vector field are lined out systematically beyond the decoupling limit, revealing a finite series justifiable within four-dimensional spacetime constraints, as highlighted by the application of the Cayley-Hamilton theorem. This theorem underpins the finite nature of the interaction terms by ensuring that certain proposed high-order terms inherently cancel out or trivialize in four dimensions. The articulation of interactions in a determinantal form parallels advancements seen in Born-Infeld electromagnetism, signifying a potential extension of this well-known theory to vector fields.

Extensions to Curved Spacetime

The work extends its applicability by generalizing vector field interactions to curved spacetimes, cognizant of preserving the critical features of the original flat spacetime interactions. To ensure only three polarizations propagate, as would be typical in Proca fields, necessary counter-interaction terms are identified to balance the gravitational couplings, drawing inspiration from Horndeski theories. This maintains the consistency of dynamical equations even when non-trivial spacetime curvature is considered.

Implications and Future Directions

This paper offers significant implications for constructing stable vector-tensor theories by delineating how derivative self-interactions can be structured to respect the foundational tenets of Proca fields. The theoretical constructs laid out provide a potential basis for exploring other extensions, such as non-linear dynamics in vector field theories and extending such ideas to multi-dimensional theories where additional higher-order interactions may be viable.

Looking forward, an interesting avenue for research would involve exploring the technical naturalness of these generalized Proca interactions concerning renormalization and quantum corrections, akin to the non-renormalization properties of scalar Galileons. Such work would bolster the robustness and applicability of these theoretical frameworks within both classical and quantum domains, deepening our understanding of massive vector fields and their role in modified theories of gravity and beyond.